276 XIV INTEGRATION IN LOCALLY COMPACT GROUPS (ii) If the sequences (A, \i) and (|A| * |/i|, v) are convolvable > then so is (A, ju, v). Likewise if the sequences (X v) and (A, |/x| * |v|) are convolvable. We may restrict ourselves to the case in which A, ^, and v are positive. Suppose that the sequence (A, /*, v) is convolvable; then for every compact subset K of G the set of triples (x, y, z) such that xyz e K is (A ® ^ ® v)- integrable. Let A be the set of pairs (x, y) such that xy e K. For each compact subset K' of G, the set A x K' c: G3 is contained in the set of triples (x, y, z) such that xyz e KK', and since KK' is compact, it follows that A x K' is ((A ® ju) ® v)-integrable. Since v ^ 0, this implies that A is (A ® ^)-integrable (13.21.11) and hence that (A, //) is convolvable. Consequently, for any f e JT+(G), it follows from the hypothesis and the Lebesgue-Fubini theorem that f *dv(z) (*f(tz) d(X * A0(0 = f *dv(z) since the function t\-*f(tz) is in Jf+(G), for each x e G. This shows that A * ^ and v are convolvable. One proves in the same way that (/*, v) and (A, n * v) are convolvable. The formula (184.108.40.206) is then a consequence of the Lebesgue- Fubini theorem. Conversely, suppose that (A, //) and (A * /*, v) are convolvable, and let / be a function belonging to «#"+(G). For each zeG, the function t\-*f(tz) belongs to Jf*4.(G), hence is (A * ^)-integrable, and we have = ||/(xyz)^0 Hence, by Lebesgue-Fubini, it follows that ** ) f which proves that (A, ju, v) is convolvable. One can give examples of measures A, ju, v on G such that the pairs (A, /*), (A * fi9 v), (fi, v) and (A, /* * v) are convolvable but (A * /*) * v ^ A * (/i * v) (Problem 1).G.